
Compute macroscopic quantities of \({\bf f}\) . We compute the discrete macroscopic charge, current and kinetic energy density \({\rho }\) , \({\bf J}\)  and \({\kappa } \in \mathbb {R}^{N_x}\)  by quadrature
\(\left(\begin{array}{l}{\rho }\\{\bf J}\\{\kappa }\end{array}\right)= \sum _{l=1}^{r} C_l\left\langle {\bf U}^{(2)}_{l},\left(\begin{array}{l}{\bf 1}_v \\{\bf v}\\\frac{1}{2}{\bf v}^2\end{array}\right)\right\rangle \ {\bf U}^{(1)}_l.\) 

Scale. We scale \({\bf f}\)  as
\(\tilde{\bf f} = \frac{1}{{\bf w}} \star {\bf f} = \sum _{l=1}^{r} \left(C_l \ \ {\bf U}_l^{(1)} \otimes \left(\frac{1}{{\bf w}} \star {\bf U}_l^{(2)}\right)\right),\) 
where \(\star \)  is the element-wise product in the \(v\) -dimension.

Project. We perform an orthogonal projection of \(\tilde{\bf f}\)  with respect to the inner product (REF ) onto subspace \(\mathcal {N}\) , i.e.
\(\langle P_{\mathcal {N}}(\tilde{\bf f}), {\bf g} \rangle _{\bf w}= \langle \tilde{\bf f}, {\bf g} \rangle _{\bf w},\quad \forall {\bf g}\in \mathcal {N}.\) 
It can be shown that \({\bf w}\star P_{\mathcal {N}}(\tilde{\bf f})\)  preserves the mass, momentum and kinetic energy densities of \({\bf f}\)  in the discrete sense.
With the orthogonal project, a conservative decomposition of \({\bf f}\)  [1]} can be performed as
\({\bf f} = {\bf w} \star (P_{\mathcal {N}}(\tilde{{\bf f}}) + (I-P_{\mathcal {N}})(\tilde{{\bf f}}))\doteq {\bf w} \star (\tilde{{\bf f}}_1 + \tilde{{\bf f}}_2)\doteq {\bf f}_1 + {\bf f}_2,\) 
where \(f_1\)  can be represented as a rank three tensor
\({\bf f}_1 ({\rho }, {\bf J}, {\kappa })= & \frac{\rho }{\Vert {\bf 1}_v\Vert _{\bf w}^2} \otimes ({\bf w} \star {\bf 1}_v)+ \frac{\bf J}{\Vert {\bf v}\Vert _{\bf w}^2} \otimes ({\bf w} \star {\bf v}) + \frac{2 {\kappa }-c{\rho }}{\Vert {\bf v}^2- c {\bf 1}_v\Vert _{\bf w}^2} \otimes ({\bf w} \star ({\bf v}^2- c {\bf 1}_v) ),\) 

where \(c=\frac{\langle \mathbf {1}_v, {\bf v}^2\rangle _{\bf w}}{\Vert {\bf 1}_v\Vert _{\bf w}^2}\)  is computed so that \(\lbrace  {\bf 1}_v, {\bf v}, {\bf v}^2- c {\bf 1}_v\rbrace \)  forms an orthogonal set of basis and
\({\rho }\) , \({\bf J}\)  and \({\kappa }\)  are the discrete mass, momentum and kinetic energy density of \({\bf f}\)  from (REF ). \({\bf f}_1\)  preserves the discrete mass, momentum and kinetic energy density of \({\bf f}\) , while the remainder part \({\bf f}_2 = {\bf f} -{\bf f}_1\)  has zero of them.

Truncate in \(l^2_{\bf w}\) . We then perform an SVD truncation of the remainder part \(\tilde{\bf f}_2\)  from (REF ) with respect to the weighted inner product (REF ). With the scaling and rescaling by the weight function, the weighted SVD truncation writes
\(\mathcal {T}^{\bf w}_{\varepsilon } (\tilde{\bf f}_2) =\frac{1}{\sqrt{\bf w}} \star \mathcal {T}_\varepsilon (\sqrt{\bf w} \star \tilde{\bf f}_2).\) 
That is, \({\bf f}_2\)  is truncated to
\({\bf w}\star \mathcal {T}^{\bf w}_{\varepsilon } (\tilde{\bf f}_2) = \sqrt{\bf w} \star \mathcal {T}_\varepsilon (\sqrt{\bf w} \star \tilde{\bf f}_2)= \sqrt{\bf w} \star \mathcal {T}_\varepsilon (\frac{1}{\sqrt{\bf w}} \star {\bf f}_2).\) 

Update. We obtain the low rank truncation of \({\bf f}\)  with local mass, momentum and energy conservation, denoted as
\(T_c({\bf f}) = {\bf f}_1 + {\bf w}\star \mathcal {T}^{\bf w}_{\varepsilon } (\tilde{\bf f}_2) ={\bf f}_1 +\sqrt{\bf w} \star \mathcal {T}_\varepsilon (\frac{1}{\sqrt{\bf w}} \star {\bf f}_2).\) 
We call the proposed truncation (REF ) the conservative truncation, as \(T_c({\bf f})\)  exactly preserves the mass, momentum and kinetic energy density of \({\bf f}\) .

